Contributions to mathematics

Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis (CH), nor the axiom of choice, can be proved from the standard Zermelo–Fraenkel axioms (ZF) of set theory. In conjunction with the earlier work of Gödel, this showed that both of these statements are logically independent of the ZF axioms: these statements can be neither proved nor disproved from these axioms. In this sense, the continuum hypothesis is undecidable, and it is the most widely known example of a natural statement that is independent from the standard ZF axioms of set theory.

For his result on the continuum hypothesis, Cohen won the Fields Medal in mathematics in 1966, and also the National Medal of Science in 1967.[5] The Fields Medal that Cohen won continues to be the only Fields Medal to be awarded for a work in mathematical logic, as of 2014.

Angus MacIntyre of the University of London stated about Cohen: "He was dauntingly clever, and one would have had to be naive or exceptionally altruistic to put one's 'hardest problem' to the Paul I knew in the '60s." He went on to compare Cohen to Kurt Gödel, saying: "Nothing more dramatic than their work has happened in the history of the subject."[6] Gödel himself wrote a letter to Cohen in 1963, a draft of which stated, "Let me repeat that it is really a delight to read your proof of the ind[ependence] of the cont[inuum] hyp[othesis]. I think that in all essential respects you have given the best possible proof & this does not happen frequently. Reading your proof had a similarly pleasant effect on me as seeing a really good play."[7]

On the continuum hypothesis

While studying the continuum hypothesis, Cohen is quoted as saying in 1985 that he had "had the feeling that people thought the problem was hopeless, since there was no new way of constructing models of set theory. Indeed, they thought you had to be slightly crazy even to think about the problem."[8]

"A point of view which the author [Cohen] feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the axiom of infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now ℵ1{\displaystyle \aleph _{1}} is the cardinality of the set of countable ordinals, and this is merely a special and the simplest way of generating a higher cardinal. The set C{\displaystyle C} [the continuum] is, in contrast, generated by a totally new and more powerful principle, namely the power set axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the replacement axiom can ever reach C{\displaystyle C}.

An "enduring and powerful product" of Cohen's work on the continuum hypothesis, and one that has been used by "countless mathematicians"[8] is known as "forcing", and it is used to construct mathematical models to test a given hypothesis for truth or falsehood.

Shortly before his death, Cohen gave a lecture describing his solution to the problem of the continuum hypothesis at the Gödel centennial conference, in Vienna in 2006. A video of this lecture is now available online.[10]